$|\mathbb{R}^\mathbb{R}|$ vs $|P(\mathbb{R})|$ Which is bigger?

Question

$|\mathbb{R}^\mathbb{R}|$ vs $|P(\mathbb{R})|$

where $\mathbb{R}^\mathbb{R} =\{f | f:\mathbb{R} \rightarrow \mathbb{R}\}$

Are they equal? Which is bigger? How can I prove it?

Answer 1

Hint: They are equal. To prove this note that $\Bbb{R^R}\subseteq\mathcal P(\Bbb{R\times R})$, and that $\Bbb{|R\times R|=|R|}$.

Answer 2

I think this can be done with a little cardinals' arithmetic:

$$\begin{align*}\left|\;\Bbb R^{\Bbb R}\;\right|&=\left(2^{\aleph_0}\right)^{2^{\aleph_0}}=2^{\aleph_0\cdot 2^{\aleph_0}}=2^{2^{\aleph_0}}\\{}\\ |P(\Bbb R)|&=2^{2^{\aleph_0}}\end{align*}\;$$

Answer 3

They are equal: $|P(\mathbb R)|=|2^{\mathbb R}|\le|\mathbb R^{\mathbb R}|\le|(2^{\mathbb R})^{\mathbb R}|=|2^{\mathbb R\times\mathbb R}|=|2^{\mathbb R}|=|P(\mathbb R)|$.